Questions: According to one study, the average shoe size of males are normally distributed with mean 10.5 and standard deviation 1.36. Sometimes if you have a really small or large foot, it can be hard to find shoes. a. What shoe size would put you in the lowest 5% of shoe sizes for males? Show some work. Round to 2 decimal places. b. Find the probability that a male had a shoe size that is 14.5 or larger. Show some work. Round answer to 4 decimal places.

Solution Steps

To find the shoe size that puts a male in the lowest \(5\%\) of shoe sizes, we need to determine the \(5^{th}\) percentile of the normal distribution with mean \(\mu = 10.5\) and standard deviation \(\sigma = 1.36\).

Using the z-score for the \(5^{th}\) percentile, we have:

\[ z_{0.05} \approx -1.6449 \]

The corresponding shoe size \(X\) can be calculated using the formula:

\[ X = \mu + z \cdot \sigma \]

Substituting the values:

\[ X = 10.5 + (-1.6449) \cdot 1.36 \approx 8.26 \]

Thus, the shoe size for the lowest \(5\%\) is:

\[ \boxed{8.26} \]

Next, we need to find the probability that a male has a shoe size of \(14.5\) or larger. This can be expressed as:

\[ P(X \geq 14.5) \]

First, we calculate the z-score for \(X = 14.5\):

\[ z = \frac{X - \mu}{\sigma} = \frac{14.5 - 10.5}{1.36} \approx 2.9412 \]

Next, we find the probability using the cumulative distribution function \(\Phi\):

\[ P(X \geq 14.5) = 1 - \Phi(z) \]

Using the z-score calculated:

\[ P(X \geq 14.5) = 1 - \Phi(2.9412) \approx 1 - 0.9984 = 0.0016 \]

Thus, the probability that a male has a shoe size of \(14.5\) or larger is:

\[ \boxed{0.0016} \]

Final Answer